Detailed Look

A more detailed look at spatial interpolation, integration through time, and I/O. For additional documentation e.g. see 1, 2, 3, 4. Here we illustrate a few things in more detail:

• reading velocities from file.
  • gridded velocity output (Udata, Vdata)
  • pre-computed trajectory output (float_traj*data)
• interpolating U,V from gridded output to individual locations
  • compared with u,v from float_traj*data
• computing trajectories (location v time) using OrdinaryDiffEq.jl
  • compared with x(t),y(t) from float_traj*data

1. Import Software

using IndividualDisplacements, OrdinaryDiffEq, DataFrames, MITgcmTools
p=dirname(pathof(IndividualDisplacements))
include(joinpath(p,"../examples/recipes_plots.jl"))
include(joinpath(p,"../examples/example123.jl"))
include(joinpath(p,"../examples/helper_functions.jl"));

2. Read Trajectory Output

from MITgcm/pkg/flt

dirIn=IndividualDisplacements.flt_example
prec=Float32
df=read_flt(dirIn,prec);

plt=plot_paths(df,300,100000.0)

3. Read Gridded Variables

using MeshArrays.jl and e.g. a NamedTyple

𝑃,Γ=example2_setup();

4. Visualize Velocity Fields

plt=heatmap(Γ["mskW"][1,1].*𝑃.u0,title="U at the start")
plt=heatmap(Γ["mskW"][1,1].*𝑃.u1-𝑃.u0,title="U end - U start")

5. Visualize Trajectories

(select one trajectory)

tmp=df[df.ID .== 200, :]
tmp[1:4,:]

4 rows × 14 columns

Super-impose trajectory over velocity field (first for u ...)

x=Γ["XG"].f[1][:,1]
y=Γ["YC"].f[1][1,:]
z=transpose(Γ["mskW"][1].*𝑃.u0);

plt=contourf(x,y,z,c=:delta) plot!(tmp[:,:lon],tmp[:,:lat],c=:red,w=4,leg=false)

Super-impose trajectory over velocity field (... then for v)

x=Γ["XC"].f[1][:,1]
y=Γ["YG"].f[1][1,:]
z=transpose(Γ["mskW"][1].*𝑃.v0);

plt=contourf(x,y,z,c=:delta) plot!(tmp[:,:lon],tmp[:,:lat],c=:red,w=4,leg=false)

6. Interpolate Velocities

dx=Γ["dx"]
uInit=[tmp[1,:lon];tmp[1,:lat]]./dx
nSteps=Int32(tmp[end,:time]/3600)-2
du=fill(0.0,2);

Visualize and compare with actual grid point values – jumps on the tangential component are expected with linear scheme:

tmpu=fill(0.0,100)
tmpv=fill(0.0,100)
tmpx=fill(0.0,100)
for i=1:100
    tmpx[i]=500.0 *i./dx
    dxdt!(du,[tmpx[i];0.499./dx],𝑃,0.0)
    tmpu[i]=du[1]
    tmpv[i]=du[2]
end

plt=plot(tmpx,tmpu,label="u (interp)")
plot!(Γ["XG"].f[1][1:10,1]./dx,𝑃.u0[1:10,1],marker=:o,label="u (C-grid)")
plot!(tmpx,tmpv,label="v (interp)")
plot!(Γ["XG"].f[1][1:10,1]./dx,𝑃.v0[1:10,1],marker=:o,label="v (C-grid)")

And similarly in the other direction

tmpu=fill(0.0,100)
tmpv=fill(0.0,100)
tmpy=fill(0.0,100)
for i=1:100
    tmpy[i]=500.0 *i./dx
    dxdt!(du,[0.499./dx;tmpy[i]],𝑃,0.0)
    tmpu[i]=du[1]
    tmpv[i]=du[2]
end

plt=plot(tmpx,tmpu,label="u (interp)") plot!(Γ["YG"].f[1][1,1:10]./dx,𝑃.u0[1,1:10],marker=:o,label="u (C-grid)") plot!(tmpx,tmpv,label="v (interp)") plot!(Γ["YG"].f[1][1,1:10]./dx,𝑃.v0[1,1:10],marker=:o,label="v (C-grid)")

Compare recomputed velocities with those from pkg/flt

nSteps=2998
tmpu=fill(0.0,nSteps); tmpv=fill(0.0,nSteps);
tmpx=fill(0.0,nSteps); tmpy=fill(0.0,nSteps);
refu=fill(0.0,nSteps); refv=fill(0.0,nSteps);
for i=1:nSteps
    dxy_dt_replay(du,[tmp[i,:lon],tmp[i,:lat]],tmp,tmp[i,:time])
    refu[i]=du[1]./dx
    refv[i]=du[2]./dx
    dxdt!(du,[tmp[i,:lon],tmp[i,:lat]]./dx,𝑃,Float64(tmp[i,:time]))
    tmpu[i]=du[1]
    tmpv[i]=du[2]
end

plt=plot(tmpu,label="u") plot!(tmpv,label="v") plot!(refu,label="u (ref)") plot!(refv,label="v (ref)")

6. Compute Trajectories

Solve through time using OrdinaryDiffEq.jl with

• dxdt! is the function computing d(position)/dt
• uInit is the initial condition u @ tspan[1]
• tspan is the time interval
• 𝑃 are parameters for dxdt!
• Tsit5 is the time-stepping scheme
• reltol and abstol are tolerance parameters

tspan = (0.0,nSteps*3600.0)
#prob = ODEProblem(dxy_dt_replay,uInit,tspan,tmp)
prob = ODEProblem(dxdt!,uInit,tspan,𝑃)
sol = solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8)
sol[1:4]

retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 4-element Array{Float64,1}:
  0.0
  0.2007818920948153
  2.2086008130429686
 22.2867900225245
u: 4-element Array{Array{Float64,1},1}:
 [37.525990625, 4.22379453125]
 [37.52598925375626, 4.223794946058778]
 [37.52597554129228, 4.223799094139329]
 [37.5258384139952, 4.223840574222589]

Compare recomputed trajectories with originals from MITgcm/pkg/flt

ref=transpose([tmp[1:nSteps,:lon] tmp[1:nSteps,:lat]])
maxLon=80*5.e3
maxLat=42*5.e3
#show(size(ref))
for i=1:nSteps-1
    ref[1,i+1]-ref[1,i]>maxLon/2 ? ref[1,i+1:end]-=fill(maxLon,(nSteps-i)) : nothing
    ref[1,i+1]-ref[1,i]<-maxLon/2 ? ref[1,i+1:end]+=fill(maxLon,(nSteps-i)) : nothing
    ref[2,i+1]-ref[2,i]>maxLat/2 ? ref[2,i+1:end]-=fill(maxLat,(nSteps-i)) : nothing
    ref[2,i+1]-ref[2,i]<-maxLat/2 ? ref[2,i+1:end]+=fill(maxLat,(nSteps-i)) : nothing
end
ref=ref./dx;

plt=plot(sol[1,:],sol[2,:],linewidth=5,title="Using Recomputed Velocities",
     xaxis="lon",yaxis="lat",label="Julia Solution") # legend=false
plot!(ref[1,:],ref[2,:],lw=3,ls=:dash,label="MITgcm Solution")

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